Simulation of Elliptical Vibration Cutting Process with Thin Shear Plane Model

Transcription

1 Simulation of Elliptical Vibation Cutting Pocess with Thin Shea Plane Model Eiji Shamoto 1, Noikazu Suzuki 1 and Rei Hino 1 1 Depatment of Mechanical Science and Engineeing Nagoa Univesit Nagoa, Aichi, Japan INTRODUCTION The elliptical vibation cutting was fist poposed in 1993 [4-7]. Since then, some ultasonic elliptical vibation tools have been developed, and thei machining pefomance has been investigated fo some difficult-to-cut mateials [8, 9,13]. Recentl, this technolog has been tested in industies mainl fo ultapecision diamond cutting of dies, molds, and optical pats. On the othe hand, thee have been less investigations on basic mechanics of the elliptical vibation cutting. An analtical model of the othogonal elliptical vibation cutting pocess, in which all velocities, foces and vibations ae pependicula to the cutting edge, was developed and veified expeimentall to pedict theoetical oughness, citical elief angle, shea angle, etc., and to claif effects of speed atio, i.e. maximum vibation speed in the nominal cutting diection / the nominal cutting speed, vetical vibation amplitude and phase diffeence between the two diectional vibations [7]. Howeve, the pacticall applied pocesses ae not geneall the othogonal tpe. The elliptical vibation locus is usuall inclined aound the cutting diection in most of the industial applications. Theefoe, simple analtical models of the thee dimensional (3D) elliptical vibation cutting pocesses ae poposed in the pesent eseach in ode to simulate the pocess quickl and to undestand the basic mechanics. The esults simulated with the pesent models ae compaed with expeimental data published in the pevious papes [7,1]. ELLIPTICAL VIBRATION CUTTING PROCESS WITH UNIQUE PHENOMENON OF FRICTION Figue 1 shows a schematic illustation of the elliptical vibation cutting pocess. The cutting stats at the time t 1, and the small pat left in the pevious ccle is cut at small depth until the time t, when the ake face contacts with the chip fomed in the pevious ccle. In the actual pocess, this small pat ma be bunished and pushed into the wokpiece and the chip because of cutting edge oundness. Afte the time t, the wokpiece mateial is sheaed and emoved as the chip at lage depth of cut. As the tangent of cutting path exceeds the shea diection at the time t 6, the fictional diection is evesed. In the actual pocess, the tool moves togethe with the chip in the shea diection nea t 6 without the sudden evesal of fiction because of elasticit. Then, the tool is sepaated fom the chip at the time t 5, when the tangent is paallel to the ake face. The educed oevesed fiction leads to significant eduction of cutting foce, eneg and heat geneation [4-6]. z x t 5 Wokpiece t 6 t 4 t t 3 θ 1 t 0 : t=0 t 1 : Beginning of cutting t : Contact betweeen ake face and chip t 3 : Bottom of locus t 4 : End of finished suface geneation t 1 t 0 α Locus of cutting edge R th t 6 : Revesal of fiction t 5 : End of cutting θ 1 : Citical elief angle R th : Theoetical ougthness FIGURE 1. Elliptical vibation cutting pocess. Figue shows the chip fomation pocesses of othogonal odina cutting and elliptical vibation cutting. It shows that the chip thickness is much smalle in the elliptical vibation cutting though the cutting conditions ae the same. Figue 3 shows the cutting foces measued in the both pocesses. The negative thust foce shown in Fig. 3(b) indicates the evesal of fiction, and this leads to the significant incease of shea angle, see Fig., and the significant

2 eduction of cutting foce, see Fig. 3 [4-6]. Figue 4 shows tpical chips obseved in pactical ultapecision elliptical vibation cutting with single cstal diamond tools. The chips ae usuall continuous and the chip thickness is almost constant as shown in the figues, although the instantaneous cutting diection is vaied. This suggests that the shea defomation ma be caused b aveage o total foce applied to the wokpiece mateial in one ccle of the elliptical vibation. (a) Odina cutting (a) Odina cutting (b) Elliptical vib. cutting [Conditions] Wokpiece: Coppe, ake: 0 deg, cutting speed: 0.6 mm/min, depth of cut: 10 μm, elliptical vibation: cicula, amp.: 10 μm p-p, feq.:1. Hz FIGURE. Diect obsevation of chip fomation pocesses. As mentioned above, the elliptical cutting pocess can be divided into the two pocesses, i.e. the mico cutting o bunishing pocess fom t 1 to t and the cutting pocess with the educed oevesed fiction fom t to t 5. When the depth of cut is lage enough elativel to the vetical amplitude of vibation, the fome mico machining peiod is negligible. On the othe hand, when the depth is extemel small and the cutting edge is shap enough, the latte cutting peiod vanishes, and the pocess ma be simila to mico milling except that the ake face is not inclined duing cutting and that the tool path is elliptical. Judging fom the obsevations of pocess and chips, it is consideed that the constant continuous chip fomation with the unique fictional phenomenon is the main pocess in man of the pactical elliptical vibation cutting. Wheeas the mechanism of the fome mico machining can be undestood as the odina mico machining pocess, the latte chip fomation pocess with the evesed oeduced fiction is unique and academicall inteesting. Theefoe, the fome pocess is not consideed and the latte pocess is modeled and discussed in the pesent eseach. (b) Elliptical vibation cutting [Conditions] Same as Fig. FIGURE 3. Tansient cutting foces measued in odina cutting and elliptical vibation cutting. [Conditions] Wokpiece: hadened steel, tool: R1 mm, ake: 0 deg, cutting speed:.5 m/min, depth of cut: 10 μm, feed ate: 10 μm/ev, vibation: cicula, amp.: 10 μm p-p, feq.:0 khz FIGURE 4. s fomed b ultapecision elliptical vibation cutting. Fo simplicit, the cutting edge is assumed shap enough, and the elastic defomation is neglected. The instantaneous depth of cut is deceased gaduall duing the latte cutting peiod, but this effect is also neglected, i.e. the depth is assumed constant. The simple thin shea plane model is emploed, and the maximum shea stess and minimum eneg pinciples ae sepaatel applied fo simple simulations and bette compehension of the continuous chip fomation pocess. The above issues neglected hee ma be consideed in the

3 futue fo moe pecise analsis and compehension. GEOMETRY OF 3D ELLITPTICAL VIBRATION CUTTING PROCESS Figue 5 shows vaious tpes of elliptical vibation cutting. Figue 5(a) is the othogonal tpe. Figue 5(b) is the oblique tpe which is intemittent cutting with sidewise motion of the cutting edge o a ota tool on aveage, which is equivalent to oblique cutting with educed width of cut [10-1]. In this case, the fiction is not educed, but the cutting foce is educed b the eduction of actual width of cut [11]. This is wh knives ae pulled in the cutting edge diection to educe the foce. actual width of cut is educed. This intemediate tpe is utilized widel in pactice. Futhemoe, the cutting edge is not alwas pependicula to the cutting diection, as shown in Fig. 5(d), whee the obliquit of cutting diection is denoted b an inclination angle i. In this case, the inclination of cutting diection enhances obliquit of the inclined elliptical vibation. The tool is vibated at an angula fequenc ω and fed at a nominal cutting speed V c, and the wokpiece is stationa hee. Then, the tool path with the vibation can be expessed as follows. x( t) = a cos( ωt) Vct cosi ( t) = bcos( ωt + ϕ ) z( t) = ccos( ωt + ϕ z ) + Vct sin i (1) Elliptical vibation Wokpiece (a)othogonal tpe z i (b) Oblique tpe x whee x, and z ae Catesian coodinates of the elative position between the tool and the wokpiece in the diection nomal to the cutting edge, the thust diection and the cutting edge diection espectivel, see Fig. 1 and Fig. 5(d). The nominal cutting diection lies on the x-z plane, i.e. finished suface. a, b and c ae vibation amplitudes in the x, and z diections espectivel. t is time and ϕ and ϕ z ae phase shifts of the vibations in the and z diections espectivel. When the minimum elative speed in the diection nomal to the ake face is negative, the cutting pocess becomes intemittent. Vc cosicosα ω ( acosα bsinα cosϕ ) + ( bsinα sinϕ ) < 0 () (c) Intemediate tpe (d) With inclination FIGURE 5. Vaious tpes of elliptical vibation cutting pocesses. Figue 5(c) shows intemediate tpe between the above two tpes, which has both effects, i.e. the thust foce is educed oevesed and the w z d x whee α is the nomal ake angle which is the ake angle in the plane nomal to the cutting edge. Unde this intemittent condition, the time t 5 (see Fig. 1) is obtained b solving Eq. (3), since the elative speed in the diection nomal to the ake face becomes zeo at t 5. { aω sin( ωt ) + V cosi} 5 c cosα bω sin( ωt5 + ϕ x )sinα = 0 (3) The ake face is sepaated fom the chip at t 5, and then it comes back to the same plane at t +T, whee T is a vibation peiod. Thus, t can be solved numeicall b the following equation.

4 { x ( t + T ) x( t5) } cosα + { ( t5) ( t + T )} sinα = 0 a{ cos( ωt ) cos( ωt )} cosα + b { cos( ωt + ϕ ) cos( ωt + ϕ )} 5 = Vc ( T t5 + t)cosα cosi 5 sinα (4) The othe paametes t 1, t 3, t 4, θ 1 and R th can also be solved in a simila wa to the othogonal elliptical vibation cutting [4]. PREDICTION OF SHEAR DIRECTION IN 3D ELLITPTICAL VIBRATION CUTTING PROCESS The shea diection in the 3D vibation cutting pocess is assumed to be constant, judging fom the diect obsevation of the pocess and the chips obtained in the pactical cutting opeations as mentioned above. The shea diection is pedicted b extending the 3D thin shea plane model [10,11] as follows. As the shea diection v s (unit vecto) is constant, the chip also moves in the same diection independentl of the instantaneous tool velocit V T, i.e. instantaneous cutting velocit. Thus, the instantaneous velocit of the chip mateial V M is deived b: VT n VM = VM vs = vs (5) vs n whee n is a unit vecto nomal to the ake face and given b ( cosα, sinα, 0), and V T is given b the time-deivative of tool position: VT = ( aω sin( ωt) Vc cosi, bω sin( ωt + ϕ ), cω sin( ωt + ϕ z ) + Vc sin i) (6) The fiction f (unit vecto) acts in the opposite diection of the elative motion between the tool and the chip. VT V f = M (7) VT VM Then, the diection of esultant cutting foce (unit vecto) can be obtained as follows, assuming a constant fictional angle β. = n cos β + f sin β (8) The esultant foce R needs to poduce the shea foce F s in the shea plane, and hence it can be calculated b: F wd v e v e s τ ( s ) + ( s x ) R = = (9) vs ( vs e )( vs )cosi whee τ is shea stess in the shea plane, w and d ae the width and depth of cut, see Fig. 5(d), and e x and e ae unit vectos in the x and diections espectivel. This foce is vaied with the vaing cutting velocit V T, and thus the instantaneous cutting powe U is given b: U = R V T (10) The shea diection is pedicted with the two fundamental pinciples sepaatel. One is the maximum shea stess pinciple. It is assumed that the shea occus in the maximum shea stess diection, whee the esultant foce makes 45 degees with the shea diection and also the shea plane [10,11]. In the pesent 3D vibation cutting pocess, the esultant foce is vaied. Theefoe, it is assumed that the shea diection is detemined b summation of the instantaneous esultant foce R sum duing the cutting peiod fom t to t 5. Rsum = V R M (11) whee V M is multiplied in the pesent pape as a weight function, because the influence on the shea defomation ma be inceased as the emoved mateial is inceased. As a esult, the shea diection v s and the summed foce R sum need to satisf the following equation. Rsum = vs cos45 + ns sin 45 Rsum (1) whee n s is a unit vecto nomal to the shea plane and given b: vs e ns = (13) vs e

5 Theefoe, the coect shea diection v s can be seached as follows. Fist, R sum is calculated b Eqs. (5-11), and then v s is checked b Eqs. (1, 13). If the eo between the left and ight sides of Eq. (1) is not small enough, the shea diection v s is updated. The coect vecto is seached numeicall b the iteation. The othe fundamental shea angle pediction is based on the minimum eneg pinciple. The shea diection is detemined hee so that the total cutting eneg E becomes minimum, whee E is given b integating instantaneous cutting powe numeicall fom t to t 5 as follows. E = UΔt (14) Thus, solution of the shea diection v s found b seaching v s can be iteativel so that the eneg becomes the minimum. The above analtical models ae applied to simulation of the 3D elliptical vibation cutting pocess hee. Howeve, the models ae also valid fo othe cutting pocesses including the 3D linea vibation cutting whee constant continuous chips ae fomed. The pesent models give the same solutions as the oblique cutting models [10,11] when i 0 and a = b = c = 0, and the ield to Mechant s [] and Kstof s [1] (o Lee-Shaffe s [3]) models when i is also zeo. Shea angle deg Maximum shea stess pinciple Minimum eneg pinciple Expeiment Speed atio [Conditions] Same as Fig. FIGURE 6. Shea angles at vaious speed atios in othogonal tpe of elliptical vibation cutting. SIMULATION RESULTS AND DISCUSSIONS The developed analtical models ae applied to the othogonal and oblique tpes of elliptical vibation cutting, and simulated esults ae compaed hee with some published data. Figue 6 shows the shea angles pedicted and measued [7] at vaious speed atios in the othogonal tpe. The fiction angle β utilized in the simulation is identified 3.5 deg b the foces measued in the odina cutting, see Fig. 3(a). Figue 7 shows the chip flow angles pedicted and measued [1] at vaious speed atios in the oblique tpe of elliptical vibation cutting. β is 41.3 deg [1] in the simulation. flow angle deg Maximum shea stess pinciple Minimum eneg pinciple Expeiment Speed atio [Conditions] Wokpiece: aluminum, ake: 0 deg, depth of cut: 50 μm, vibation: cicula, amp.: 1 mm p-p, feq.:10 Hz FIGURE 7. flow angles at vaious speed atios in oblique tpe of elliptical vibation cutting. Consideing the simplicit of analtical models, the chaacteistic angles pedicted with eithe pinciple agee oughl with the expeimental data, as shown in the figues. It shows that the two tpes of elliptical vibation cutting pocess can be pedicted quickl and oughl with the models developed hee. The ough ageements suggest that the othogonal and oblique tpes of elliptical vibation cutting can be undestood as the unique chip fomation pocess with evesal of fiction and the oblique cutting with the educed width of cut espectivel. Tansient cutting foces in the intemediate tpe ae also pedicted and compaed with expeimental data [1] in Fig. 8. β is 41.3 deg and τ is 94.3 MPa [1]. The esults ae in a good ageement duing the cutting peiod, as shown in the figue. The disageement is consideed to be caused mainl b the elastic defomation, which is neglected hee. The thust foce is evesed in the cutting peiod b the othogonal component of the elliptical vibation. The foce in the cutting edge diection, which is shown b nomal in the figue, coesponds to the

6 obliquit of cutting pocess. This is caused b the oblique component of the elliptical vibation, and it also educes the cutting foce b educing the actual width of cut [11]. The measued peak pincipal foce was 140 N as shown in the figue, while aveage pincipal foce measued in the odina cutting was 480 N [1]. It is consideed that this significant eduction is caused b combination of the above two effects. Cutting foce N Maximum shea stess pinciple Minimum eneg pinciple Expeiment Pincipal Thust Nomal Time s [Conditions] Wokpiece: aluminum, ake: 0 deg, depth of cut: 0.3 mm, width of cut: mm, cutting speed: 94.3 mm/min, vibation: cicula, amp.: 1 mm p-p, feq.:10 Hz, inclination of elliptical vibation: 80 deg fom othogonal tpe, i.e. 10 deg fom finished suface FIGURE 8. Tansient cutting foces in intemediate tpe of elliptical vibation cutting. CONCLUSION The simple analtical models of the 3D elliptical vibation cutting ae developed b emploing the thin shea plane model and eithe the maximum shea stess pinciple o the minimum eneg pinciple. The simulated esults agee oughl with the expeimental data published peviousl. The models epesent the unique cutting mechanics of two tpes of elliptical vibation cutting. The simplified models and the simulated and expeimental data ae helpful to undestand the pocesses. The suggest that the continuous chip fomation pocess in the oblique tpe of elliptical vibation cutting is the unique cutting pocess with the evesal phenomenon of fiction, and the oblique tpe of elliptical vibation cutting is equivalent on aveage to the cutting pocess with the sidewise motion of tool. The pactical elliptical vibation cutting can be undestood as the intemediate pocess between the two tpes. REFERENCES [1] Kstof J. Beichte ube Betiebswissenschaftliche Abeiten, Bd., 1, VDI Velag.: 1939 [] Mechant M.E. Mechanics of the Metal Cutting Pocess. II. Plasticit Conditions in Othogonal Cutting. J. Applied Phsics: 1945; 16: [3] Lee, E.H., Shaffe B.W. The Theo of Plasticit Applied to a Poblem of Machining. J. Applied Mechanics: 1951; 18; [4] Shamoto E., Moiwaki T. Fundamental Stud on Elliptical Vibation Cutting. Poc. 8th Annual Meeting, ASPE: 1993; [5] Shamoto E., Moiwaki T. Stud on Elliptical Vibation Cutting. Ann. CIRP: 1994; 43/1: [6] Shamoto E., Moimoto Y., Moiwaki T. Elliptical Vibation Cutting (1st Repot, Cutting Pinciple and Basic Pefomance) (in Japanese). J. JSPE: 1996; 6/8: [7] Shamoto E., Moimoto Y., Moiwaki T. Elliptical Vibation Cutting (nd epot, Stud on Effects of Vibation Conditions) (in Japanese). J. JSPE: 1999; 65/3: [8] Shamoto E., Moiwaki T. Ultapecision Diamond Cutting of Hadened Steel b Appling Elliptical Vibation Cutting. Ann. CIRP: 1999; 48/1: [9] Shamoto E., Ma C.X., Moiwaki T. Ultapecision Ductile Cutting of Glass b Appling Ultasonic Elliptical Vibation Cutting. Poc. of 1st Int. Conf. and geneal meeting of EUSPEN: 1999; [10] Shamoto E., Altintas Y. Pediction of Shea Angle in Oblique Cutting with Maximum Shea Stess and Minimum Eneg Pinciples. Tans. ASME J. Manuf. Sci. Eng.: 1999; 11: [11] Shamoto E. Stud on Thee Dimensional Cutting Mechanics (1st Repot, Compehension and Vecto Fomulation of Oblique Cutting Pocess) (in Japanese). J. JSPE: 00; 68/3: [1] Shamoto E., Song Y.C., Sassa K., Yoshida H., Hino R., Moiwaki T. Poposal of Oblique Tpe of Elliptical Vibation Cutting and Its Basic Pefomance. J. JSPE: 003; 69/7: [13] Suzuki N, Haitani M, Yang J., Hino R., Shamoto E. Elliptical Vibation Cutting of Tungsten Allo Molds fo Optical Glass Pats. Ann. CIRP: 007; 56/1: to be published